In the essays Having Understanding Versus Knowing Correct Explanations and Understanding, Shallow Thinking, and Schools, I explain why knowing a teacher's explanation of some phenomenon is not necessarily to understand the phenomenon. In this essay, I want to examine the issue of whether understanding a phenomena or relatively complex subject matter can ever be the same thing as knowing any particular set of propositions about the phenomenon or subject.  That is, is understanding something ever the same thing as knowing various (particular) things about it?

The dilemma is that on the one hand it seems that understanding some subject (say understanding photography, or understanding mechanics in physics, or understanding rate-time-distance problems in math or understanding baseball) means you know some finite set of propositions about it -- propositions of the sort that appear in textbooks to explain that subject along with all the propositions good teachers might give as additional information when students raise questions or make mistakes.  Moreover, it seems that students who do gain understanding have done so by reading or hearing such propositions, perhaps along with some other propositions they have figured out for themselves as they thought about the topic, worked problems, etc.  Surely, there are not an infinite set of such propositions that people have to learn in order to understand the topic.  If a person who understands a topic tells all he knows about it, wouldn't knowing all those things then be the same thing as understanding the topic?

On the other hand, even when students learn all these propositions and can recite them, they do not always understand the topic.  For example, they may make non-trivial mistakes in working problems -- mistakes that are not just calculating errors, but mistakes in the way they think the problem needs to be worked; they do it wrong.  Or they may not know how to do the problem at all even though they have, and in some way know, all the information they would need to work the problem.  They just don't know how to use it; they cannot see how to apply it.  But since even someone who understands a topic may occasionally be stumped about how to work a particular problem or how to resolve a particular issue in the subject, what is it that makes not being able to solve a problem in the field correctly at a particular time a sign of lack of understanding rather than a temporary confusion or lack of insight?

And finally, since many students can work certain kinds of problems by rote, by following recipes mechanically, and yet not understand what they are doing (as can be demonstrated by asking conceptual questions, or as can be demonstrated by modifying the problems in ways that don't quite fit the recipes without some modifications that require some insight), the ability to solve particular problems, by itself, does not seem to be what is meant by understanding the topic. Someone who "understood" the recipes could, generally, still use them to solve the modified problem because s/he would transform the formulas or what they represent as needed, but someone who only knows the recipes and can only use them mechanically, but does not "understand" them, generally will not be able to solve problems that do not simply plug into them.

So what does it mean to understand something?

First, this is a difficult question, and I am not particularly confident of my answer to it.  But I suggest the following for consideration.

It seems to me that understanding something means "seeing" or having insight into how it "works", and that what one knows describes part of that insight, but more importantly, it derives from that insight, and is not the same thing as the insight itself.  Suppose we ask someone who "understands" baseball to tell us all he knows and that he writes down everything he can think of.  It might fill volumes.  Still, we might be sitting next to him in the dugout as he coaches a game, and he leans over and tells the manager, Jones can steal on this pitcher because his high leg kick makes his delivery be too slow to the plate.  If we ask why, if he knows this, he didn't write it down in the book, he might say it was because he never saw this pitcher before and he didn't know about the leg kick.  Plus, he knows of some pitchers with high kicks who can still get the ball to the plate fast enough, but this pitcher does not seem to be one of them.  Or it might turn out that he wrote a chapter on stealing bases and that part of that chapter talked about slow pitcher deliveries and their causes, so that in a sense he did write about this particular thing, but without being able to identify every future pitcher ahead of time who might be too slow.  He might have written about the concept in general or at a level of abstraction that encompasses this particular case.

But suppose while he is watching the game, he comes up with an idea he had never had before and did not write down in the book, but which he thinks of now simply because he sees something in a particular circumstance that, coupled with his knowledge and understanding of the game, he sees will work.  He figures out a new proposition, perhaps a particular one in this case, but perhaps later he will be able to generalize it somehow.  It seems to me that the new proposition stems from his understanding and was neither a previous part of it, nor does it, by itself enlarge his understanding of the game.  However, whatever insight he gets from it that allows him to generalize, may, and probably does, enlarge his understanding of the game.  Notice that it seems quite natural to say that "the insight allows him to generalize" rather than saying "the insight is the generalization."

When one of my children was about six or seven years old, I gave her a scrambled Rubik's Cube, showed her how it turned in various ways, and explained the object was to get its faces each to be one color.  She took it and played with it for some time and then came back to me with one face all in the same color.  I examined it and noticed, however, that along each edge of that face, the colors were all jumbled up, so that though all the yellow faces were on one side, none or only one of the mini-cubes with yellow on them had their adjacent side (e.g., red) on the side it needed to be (e.g., the red side).  I pointed that out to my daughter who took the cube and then thought about what I had said.  She then looked at me and said "You mean all the sides of all these little yellow pieces have to be on the right sides, and in the right places too?"  I said "Yes." She thought about it a moment longer and then handed the cube back to me and said "That would be hard."  She was done with it. She had had all the insight into the cube that she cared about.  She probably could not articulate the problems with doing that, but she could "see" or appreciate them from what she had observed while working on the side she had got done. 

On a more positive note, while I was working with trying to solve the cube, I had progressed to the point of getting all but the last side in place, but I didn't see how to rearrange the pieces in it without messing up the parts I already had in place.  I was pretty stuck.  Then while I was driving somewhere and thinking about the problem, a solution took shape in my mind, but it was neither in pictures nor in words.  It was something of a concept.  I thought of it as moving some of the set pieces to an area where I could sort of squirrel them away while I manipulated the pieces I needed to and then return them from storage.  I could see the concept working in general in my mind, though I could not quite see the details in a picture in my mind.  As soon as I got back to my office I tried what I thought would work, and it did.  It was something like seven steps to move each piece and put all the faces back in place up to that point to make sure and to get ready to move the next piece.  I could neither state how to do it in words, nor could I picture the precise steps in my head but I understood the concept I had figured out.  After I did it enough and became real comfortable with the sequence of manipulations, I could probably have written down the procedure as a set of directions in statement form, but that is not how it came to me. 

A comparable thing on a more mundane level is the sort of situation where someone asks you for directions to some place, and though you easily know how to get there if you were driving yourself, you never thought of spelling it out for someone or figuring out the best way to try to tell them to go.  Sometimes the way you would go would be too difficult for someone to have to follow, so you may give them a route that takes a bit longer but is easier to navigate.  Often when something like this occurs and someone asks you for directions, you might hesitate a moment, and they might then think you don't know and make a comment to that effect.  Your response will be something like "Oh, no, I know how to get there; I am just trying to figure out the best way to tell you to go."  Your knowledge is not the set of propositions you eventually come out with; you figure those out by using your knowledge of how to get there.  You have the knowledge before you have the set of propositions that serve as the set of directions. Morever, if you have driven in any city for a long time, you normally know far more places and how to get to them than any set of statements you may have ever thought about.  Knowing or understanding how to get around in the city is not just having a huge number of sets of directions as statements in your head, nor does it even mean you could tell someone how to get to a certain place because you may not remember street names or landmarks (without seeing them in front of you) or exactly what the cues are you have that let you remember at the time where to turn and where to go next.

At an even more mundane level, suppose you report that "the cat is on the mat" outside.  That statement does not say all that you know or have observed about the cat.  It does not say whether the cat is lying down, sitting, standing, or pacing.  It does not say whether it appears to be pining to come in.  It does not say that if it is lying down, whether it is on its side or on its stomach, whether it is stretched out peacefully or is attentively watching a bird or whether it seems wary and guarded.  It does not say whether it is lying north-south, east-west or somewhere between.  All the things you see are not all the things you report; and it would be very hard to report them all even in this simple situation.

We tend to report the things we think are salient or necessary to report.  We almost never even attempt a full description of anything, even when we give directions or an explanation of it.  We simply say the things we think are important and relevant.

But why then, are there situations where people might be thought or known to understand something and yet they cannot derive the proper information or set of statements to solve a problem?  Why is it that we might believe someone has great understanding of something but that some solutions will escape him/her though s/he will recognize and appreciate it immediately if someone points it out? For example, someone might understand algebra or a particular algebraic technique, but not be able to solve a particular problem correctly at a particular time.  Below there is a link to some sample problems that most people will miss even though they will have the understanding to be able to get it right, and there is one math problem in particular, which is more likely to be missed by mathematicians who have the most mathematical understanding.   So the issue here is how can we be said to understand a phenomenon if we cannot figure out some particular aspect of it at a particular time.

I think there are a number of reasons for not being able to solve particular problems or deal properly with a phenomenon which we could still be said to understand:

1) Being able to derive a solution to a problem about some phenomenon one understands is perhaps like other kinds of abilities in which having the ability does not mean you can produce a particular result each time. For example, we can perfectly accurately say that each time Mark McGwire stepped to the plate during his most productive seasons, he had the ability to hit a home run.  What that means is that he has the power to do it, and he has the reflexes and coordination to do it, and that he can do it fairly frequently, though he does not always do it, and though he actually hits home runs far fewer times than not.  It means that pitchers have to be very careful with him, much moreso than they would with someone like me, who could probably not hit a home run no matter how the ball was pitched.  And it means they have to be much more careful with him than they would with someone who can hit home runs but who seldom does.

Certain abilities, such as the ability to hit home runs, may be a frequency or relative kind of thing, based on the sorts of skills one displays, even apart from actually doing the thing one displays the ability to do.  Jack Nicklaus and many others prophecied long before Tiger Woods won any professional major tournaments that he would win more than anyone else.  As of this writing that remains to be seen, but Woods is definitely off to a good start, and he has already won the Master's twice and at least three other majors once.  Yet Woods does not win every tournament he plays nor every major tournament he plays.  Still it was clear he has the ability to win them, is often the "favorite" to have to beat going in to a tournament, and he displayed this before he even played in any professional major tournaments.  His wins have only confirmed he has this ability; they were not what (first) demonstrated it.

2) Similarly, I think that one can understand a phenomenon or subject without therefore knowing all possible ways to explain it, account for it, or demonstrate it.  Physicist Richard Feynman has a set of essays explaining why planets have eliptical orbits rather than circular ones.  He understands that phenomenon.  But what he was trying to do was to explain in ways that students could understand, Newton's explanation of the issue.  There is a point in Newton's notes however where Feynman could not understand part of Newton's derivation.  Newton relied on some fact about conic sections that was readily known at the time, but the study of which had become essentially neglected, or at least had not formed part of Feynman's intellectual background, and he could not see how Newton went from one particular step to another.  So Feynman thought about the matter and devised his own way to derive and explain the step.  The fact he could not understand Newton's step, or that he could not have the same understanding of the issue that Newton did, did not mean he did not understand the phenomenon.

3) Also, there will simply be times when one will not focus on something one needs to be attending to. In another essay, in a different context I give some examples of fairly simple problems that are designed to trick people into giving the wrong answers about issues they may perfectly well understand because the problems psychologically get you to focus on the wrong things even though in some sense you know better.  There are many such "brain teaser" problems and there are many more such things that occur in normal life quite naturally.  Understanding what is necessary to solve a particular problem at a particular time does not mean one will always be able see at that time that the problem fits the relevant things one knows.  When I first bought a close-up attachment set for my camera -- a set of lenses that allow you to photograph relatively small objects from closer than you can focus with a normal lense -- I went out in search of subject matter to try them.  I was in a vacant, uncared for field near where I lived and in it stood a fairly tall, particularly pretty Queen Anne's Lace  -- a flowering plant with a horizontal flat flower that resembles a round piece of finely made lace material.  I got it in my head that it would be neat to photograph it with the sun coming through the flower, so I got down on the ground to shoot up from under it toward the sun.  I could not get down low enough on my stomach to get the angle on the sun, so I turned over on my back, and shinnied under the plant.  The sun was still too high in the sky and the plant to short for me to get the angle I needed.  I decided I would have to come back later when the sun was lower and I did not have to be directly under the plant in order to have it be between the camera and the sun.  As I started to walk away, I realized that since this was just a weed in an unused field I could simply pluck it, and hold it up where I needed to.  I did that.  But I felt totally stupid that I had not thought of that immediately instead of spending ten minutes crawling around in the dirt trying to get sufficiently under the plant.  It was not that I did not understand I could have plucked the weed to put it above my camera; it was that I was focused on getting my camera under the plant and didn't think about moving the plant instead of my camera.

4) There are cases where one's understanding is enlarged, though, as one makes more and more derivations about a particular subject and as one is able to keep in mind (the rudiments at least) of what one has derived.  For example, when many of the best physicists in America were gathered together at Los Alamos to figure out how to make an atomic explosion, if they could, they already had an understanding of nuclear energy, moreso than non-physicists or physicists who studied other fields of physics.  But clearly their understanding of atomic energy grew as they made derivations and calculations and as they ran into problems and conceptual difficulties.  Similarly, knowing and understanding all the axioms and postulates of plane geometry (or of any mathematical field) is not the same thing as understanding geometry (or whatever field).  One has to actually make the derivations and work with them until one is familiar and comfortable with them. 

The short version is that there are 10 numerals: 0 - 9, and we can thus easily write down the numbers whose verbal names are from zero through nine.  But when we try to write "ten" in number form, we are now stuck because we have used all our numerals and we do not have any more.  We could change colors or we could write the next set of 10 numbers (from ten to nineteen) by lying the numerals on their sides, but we need to do something.  What was decided some time ago was that we should add another column, and because the first number that we need it for is ten, we could call the column the "ten's" column and let it designate how many tens there are in any number we are trying to write.  So if we are trying to write 64, that would be six groups of ten and then four ones.  In that essay we adapt the same methodology for writing numbers for a hypothetical alien civilization where the aliens all have just two fingers and (thus) just two numerals (0, and 1).  After we write zero and one, we are now stuck and thus need to make a new column to write "two" in numeral form.  We start another column and write a numeral "1" in it to designate there is one "two" in two, and we then write "0" in the original column because there is nothing besides the two in the quantity designated by the name "two".  If you go to that essay, you will see how all the numbers can be written using this two-numeral, or binary, system. 

The point that I want to make here is that if a student follows the teaching method presented in that essay, as the students in that class did, they will see not only how to write numbers in binary form, but they will see how to write numbers in any "base" form, where the base simply describes how many numerals you have at your disposal to use.  Many elementary school math teachers who have read and followed that essay have said that the essay helped them realize for the first time how our own written numbering system worked.  They say it is the first time they have "understood" how our system of writing numbers works.  These were people who have for a long time known (the facts of) how to write numbers in the decimal system, but they did not know there was anything more to it than that.  If you had simply introduced them to the written numbers in a different (base) system, training them by rote how to write the numbers as we teach kids to write numbers in our own system, they might have been able to learn that sequence, but would not have seen any correlation between it and the numbers in our system.  There is a point in going through Socratic binary essay, where the "light will dawn" on students what is going on in our own system of writing numbers or any other base system that writes numbers in the same way, but it will not be because of some specific set of propositions that describe all that is going on.

Teaching is about getting people to see what you see by using various propositions until they can see it, but the propositions you use are not themselves what you see nor what you want them to see.  The propositions are a (usually limited) way of trying to explain what you see.  And my current view is that we use all the propositions we think necessary, and that we can come up with, in order to try to get students, through thinking about the ideas those propositions involve, to gain the understanding we have.

I think this is also more than just saying that the Socratic binary math essay teaches writing numbers at a higher level of abstraction or generalization than does just teaching students how to write numbers in the decimal or the binary system.  It does teach a higher level of abstraction or generalization about writing numbers, but that just comes with the understanding; the propositions themselves are not somehow more abstract, nor is the presentation.  In fact, it is all quite simple and concrete in form.

If we look at classical physics as being expressed in its most general form by Newton's three laws of motion, that still does not help students apply those laws in each new unit.  Some people seem to have an intuitive understanding of how to apply the laws and others, like me, will invariably misapply them in trying to solve a new kind of problem.  Nobel physicist Richard Feynman had a really great intuitive sense of how things worked in regard to physics or mathematical principles, but he still had to work in many cases to expand his understanding in some new areas.  His father had taught him from an early age to think about what propositions meant in terms other than just how those propositions were stated.  For example, if his father were reading to him about a kind of dinosaur that was 25 feet tall and 80 feet long, his father would point out that entailed that the dinosaur could stand in their front yard and stoop down a bit to look in the second floor windows right near the house and its tail would be trailing into the street.

One of his more remarkable (to me) cases of understanding was during a high school math team competition when the question was something like "A rowing team is rowing upstream at 1.5 miles per hour relative to the bank, and the water is flowing downstream at 3 miles per hour relative to the bank.  The guy in the back of the boat is wearing a hat which falls into the water and floats downstream with the current.  If the rowing crew does not notice the hat is missing for 15 minutes, how long will it take them to row back and catch the hat if they row with the same force downstream that they have been rowing upstream (and if they can change direction instantaneously).  Feynman immediately said the answer was 15 minutes because he could see that the times relative to the bank had nothing to do with the problem, and that it would take the same time to catch up with the hat as it took to notice the hat was missing.  If you work the problem out algebraically, there is a place in the calculations where the rates relative to the bank upstream and downstream do cancel out, but the only way I can see that it works is to think of driving from one city to another directly west of it and then back along the same road at the same speed.  It will take the same amount of time even though the earth has turned nearly 1000 miles an hour toward the east while you were driving. The motion of the surface on which you are moving, relative to some fixed point outside that surface, does not affect the relative motions to each other of things on that surface, it seems.

In chemistry, my younger daughter was having difficulty understanding the concept of mols, which is the number of particles (atoms or molecules) that make up the atomic or molecular weight of an element or compound in grams.  I explained it to her just about every which way I could, using examples from grapes to bowling balls to just about any kind of weight-versus-quantity notion or relationship I could think of.  She could almost "see" it, but not quite.  Each time she thought she had it, I would ask a question about a some different kind of case, and she would mess it up and get the wrong answer.  She was trying, and she is intelligent, but it just wasn't quite sinking in.  It was getting funny for her and me because of the way we interact about these kinds of things and because we both know she is smarter than I am and so if I can see it, she knows she ought to be able to see it too.  She also could tell it probably wasn't that difficult and that she was really close to seeing it, but just couldn't quite get over whatever conceptual hump was keeping her from it.  At one point, partially out of excitement and partially out of frustration, but more out of trying to lighten up the situation a little, when she got it wrong again by making a particular kind of conceptual mistake, I yelled at her something like "NO! NO! NO!  you nitwit! [She knows I am kidding when I call her these kinds of names; we do that to each other when we get into one of these kinds of situations or when one of us makes an obviously understandable mistake.] You are still thinking of it as ... instead of as...!"  My wife came into the room to ask why I was yelling and calling her names, and I said calmly "I am just teaching; this concept is difficult and requires some yelling to get it across.  Don't you know anything about teaching?" (She has a PhD in education.)  The yelling was to try to get my daughter to focus on a certain aspect of what I was trying to show her, and to get her to mentally regroup in some way, in order to get her to make that intellectual leap that is between just knowing what has been said and seeing it or understanding it.  She finally got it, and then she could work all the problems and answer all my subsequent cases.  Neither she nor I could tell what the difference was between knowing and not knowing, but it was not an issue of her having learned any new propositions.  It was about her being able to see what the propositions were trying to point out.  In the introduction to the online explanation I give about "mols", I also say there that it is not just reading what I have written but it will take thinking about what I have written in order to understand the concept because the concept is actually quite strange.  In a sense it is more strange than it is complex, and it is its strangeness which makes it difficult.  Once you see the concept, however, the problems that involve it are fairly easy to work.

There are two things that this seems somewhat analogous to, but these are just loose analogies, and I don't want to imply that these are the same exact kinds of things: 

1) There is a difference between playing musical notes in a mechanical form the way they are written, and playing the music that those notes represent.  Musical notes are an approximation of the music; they are not the music.  Playing a piece of music "mechanically" by just playing the notes technically correctly is not the same thing as bringing the music to life.  In some ways, it is the difference between reading literature without much expression and reading it with great expression. 

2) But the second analogy is one that I like even better.  Many of us have had the experience of being shown an optical illusion that we cannot "make out" in the way it is being told to us.  One of the difficult ones for me is the image that shows a set of stairs both going up in one direction or going down in that same direction, depending on how you are seeing it.  Each time I look at it after not having seen it in a while, I tend to see it only one way, and have to just keep looking or closing my eyes for a while until it happens to invert for me.  Once it does that, I can often tend to go back and forth between the two images, but it is not clear how I am doing it or what I am thinking when I do it.  What does not help is for someone to say, when you cannot see it, something like "See here is the top step and here is the bottom step."  When you cannot see the whole set of steps you also normally cannot see any of the steps.  Explanations don't help.  You just have to know what you are looking for, or supposed to be seeing, and then you have to keep looking at it until it finally "appears" to you.  When it does finally appear, it will not be because of any new cognitive information, but it will be because of some change in perspective or something.  Sometimes once you see the inversion of what you were seeing before, you cannot easily get back to the first image. 

The other optical illusion that is always difficult for me when I have not seen it in a while is the one that is usually described as the image of the face of Jesus in the mud, or in essence in a bunch of black and white blobs that look like ink blots spilled all over paper.  When you can see the face, you can point out quite easily where the nose and mouth and the eyes, etc. are, but that usually does not in any way help anyone who cannot see the face in the image.  As with the inverting steps, if you cannot see --distinguish-- a face, you also cannot distinguish the parts of the face.  Someone can point to a place on the paper and say, "See, this is His left eye" and you will see nothing that all that resembles an eye even though you are looking exactly where s/he is pointing.  They can point out individual features until the cows come home, and you won't be able to see them or the whole face, until suddenly the whole thing just dawns in front of your eyes to you.  Then you won't be able to tell how you could not see it before.  Let it go for a few hours or a few weeks, and I have to start all over again because I cannot see it again.

This means when schools are trying to teach a topic for understanding, they need to do more than to try to get students to learn particular propositions.  They need to try to get students to have the insights and perspectives those propositions try to convey and from which they derive.  That is a more difficult task in many cases than is simply getting students to memorize or know a set of particular statements.  It is an art, and it takes methods for discovering what students are both "seeing" and not "seeing" about the subject, regardless of what they can recite.  

Then it takes insight into both the topic and the student's particular obstacle to seeing it, to try to devise an explanation or a technique that will help the child get past that obstacle.  Often this involves trying to unleash the student's powers of insight or to focus it on some aspect of the phenomena that is causing the stumbling block.  Again, this is an art.  Group instruction to teach understanding to individuals works when the understanding of different students is unleashed by the same things, or when the teacher says or does enough different things in teaching so that each student's power of understanding will be fostered or guided by at least one of those things.  There are some techniques, such as the Socratic Method, which seem to unleash and focus students' understanding more effectively than others.  But good lecturing can also do that in many cases.  There are lecturers who are far more effective at fostering understanding than others.  Lecturing does not need to mean simply standing before a group stating propositions which students put into their notes.  Lectures can also ask questions, raise issues, arouse cognitive dissonance by seeming to contradict themselves or by seeming to contradict what is obvious or what is conventional wisdom.  A good lecture can stimulate thinking and focus it in particular ways, not put it to sleep.  And in teaching for understanding, it is the focused stimulation of thinking that is important.  The reason knowledge, in areas that require understanding, not just factual information, is not something that can be "poured into" students by teaching them particular facts is that such knowledge simply is not those particular facts.  It is the insight into what makes those facts be true.  And it is the insight that those facts try to convey.

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I say "complex" subject because it seems to me that there are some simple subjects, such as in defining some words, in which a proposition or two will suffice to capture the meaning of the term.  For example, one might understand the meaning of the word Mrs. to mean the title one uses in addressing a married, widowed, or divorced woman by addressing her or speaking about her as "Mrs. Smith", where one uses the title and her last name.  If one understands that usage, one understands what there is to know about the word Mrs.  Some definitions are not as simple, however, as in the way physicists use the word force, or a seemingly simple word like bachelor, which is an unmarried man who has never been married but who is of an age eligible for marriage.  The concept of "an age eligible for marriage" or what determines such an age might be difficult to explain or to understand.  (Return to text.)